Let's solve the problems one by one.

1. Find the inverse of $f(x) = x^5$

To find the inverse of this function:

• Start by writing $y = x^5$.
• Solve for $x$ in terms of $y$: $y = x^5 \implies x = \sqrt[5]{y}$
• Now, switch $y$ and $x$ to express the inverse: $f^{-1}(x) = \sqrt[5]{x}$ So, the inverse is $f^{-1}(x) = \sqrt[5]{x}$.

2. Find the inverse of $f(x) = \sqrt[9]{x + 3}$

For this one:

• Write $y = \sqrt[9]{x + 3}$.
• Eliminate the radical by raising both sides to the power of 9: $y^9 = x + 3$
• Solve for $x$: $x = y^9 - 3$
• Finally, switch $y$ and $x$: $f^{-1}(x) = x^9 - 3$ So, the inverse is $f^{-1}(x) = x^9 - 3$, which matches the provided answer.

3. Find the inverse of $f(x) = (x + 1)^7 - 3$

For this function:

• Write $y = (x + 1)^7 - 3$.
• Solve for $x$. First, add 3 to both sides: $y + 3 = (x + 1)^7$
• Take the 7th root of both sides: $\sqrt[7]{y + 3} = x + 1$
• Subtract 1 from both sides: $x = \sqrt[7]{y + 3} - 1$
• Switch $y$ and $x$: $f^{-1}(x) = \sqrt[7]{x + 3} - 1$ So, the inverse is $f^{-1}(x) = \sqrt[7]{x + 3} - 1$.

Do you need further clarification or more details?

Here are 5 related questions:

1. How do you verify if two functions are inverses of each other?
2. What are the conditions for a function to have an inverse?
3. How would you find the inverse of a logarithmic function?
4. Can all polynomial functions be inverted? Why or why not?
5. How does the graph of a function compare to the graph of its inverse?

Tip: Remember, when solving for the inverse of a function, the goal is to express the dependent variable in terms of the independent variable, then swap them!